240 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF
so == S (flj cos ttj sin (</> + €1 - ft) - 61 sin ^ cos (<£ + ea - ft)} y = $ {ot sin cq sin (<£ + e^ - ft) + 6t cos at cos (<£ -}- e1 - pj},
where 8 denotes the sum of the expression written down and that formed from it by replacing the suffix 1 by 2. To form the expression for the intensity, or rather what would be the intensity if the quantities c and e were absolutely constant, not merely constant for a great number of successive undulations, we must develope the expressions for x and y so as to contain the sine and cosine of $ + K, and take the sum of the squares of the coefficients. K is here a constant quantity which may be chosen at pleasure, and which it will be convenient to take equal to e1 — ft. If / be the intensity, in the sense above explained, or as it may be called the temporary intensity, we find, putting S for e,-ft_€l + ft, J = {a1 cos ax + a2 cos a2 cos 8 4- 12 sin a2 sin S}2
4- {— &t sin tfj 4- a2 cos a2 sin S — &2 sin a2 cos S}2 + (ax sin ax -f a2 sin a2 cos S — 62 cos a2 sin S}2
4- {6j cos o^ + a2 sin a2 sin S + &2 cos a2 cos S}2
- a/ + V + < + V + 2 (aA + 6469) cos (a2 - aj cos S
+ 2 (a^ 4- aa&t) sin («2 - at) sin S.
On putting for a, 6 their values c cos /3, c sin /3, this expression becomes
1 = ci2 + ca2 + 2cA (cos (a2 - ai) cos (/32 - /3i) cos s
+ sin (a2 - otj) sin (/32 + /3X) sin 8} ... (9).
In order that / may be independent of the difference of phase p2"~ Pi> an(^ therefore of S, we must have either
cos (a, ~ ax) = 0, sin (y32 + &) = 0,......(10),
or sin (a2 - a,) = 0, cos (p9 - /3L) = 0,......(11).
The equations (10) give «2- a,= ± 90°, £2 = - £,, so that the ellipses described in the case of the two streams are similar, their major axes perpendicular to each other, and the direction of revolution in the one stream contrary to that in the other. It will be easily seen that the equations (11) differ from (10) only in this, that what are regarded as the first and second principal planes of the second stream when equations (10) are satisfied, are accounted respectively the second and first when (11) are satisfied.